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You can proceed similarly to this question. For $0 < T_0\le T$, consider the option with payoff, at the option maturity $T_0$, of the form \begin{align*} \max(F_{T_0, T}-K, \, 0).\tag{1} \end{align*} Note that \begin{align*} F_{T_0, T} &= F_{0, T}\exp\Bigg(-\frac{1}{2}\int_0^{T_0} \left[\left(h_1e^{-\lambda (T-t)}+h_0\right)^2 + h_2^2e^{-2\lambda (T-...


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Generally, the PCA approach proceeds as follows. Consider historical observation times $t_0 < t_1 < \cdots < t_K \le0$. For bucketing times $\delta_1 < \cdots \delta_n$, let $X_j(t_k) = \ln F(t_k, t_k+\delta_j)$. Moreover, let $\eta_j$, for $j=1, \ldots, n$, be a normal random variable with a sample set $\big\{X_j(t_k)-X_j(t_{k-1}\}_{k=1}^K\big\}...



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